Optimal. Leaf size=88 \[ -\frac {1}{4} \log \left (\sqrt [3]{x^2-4 x-4}+2\right )+\frac {1}{8} \log \left (\left (x^2-4 x-4\right )^{2/3}-2 \sqrt [3]{x^2-4 x-4}+4\right )-\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{x^2-4 x-4}}{\sqrt {3}}\right ) \]
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Rubi [A] time = 0.05, antiderivative size = 60, normalized size of antiderivative = 0.68, number of steps used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {694, 266, 56, 618, 204, 31} \begin {gather*} -\frac {3}{8} \log \left (\sqrt [3]{(x-2)^2-8}+2\right )+\frac {1}{4} \log (2-x)-\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1-\sqrt [3]{(x-2)^2-8}}{\sqrt {3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 31
Rule 56
Rule 204
Rule 266
Rule 618
Rule 694
Rubi steps
\begin {align*} \int \frac {1}{(-2+x) \sqrt [3]{-4-4 x+x^2}} \, dx &=\operatorname {Subst}\left (\int \frac {1}{x \sqrt [3]{-8+x^2}} \, dx,x,-2+x\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt [3]{-8+x} x} \, dx,x,(-2+x)^2\right )\\ &=\frac {1}{4} \log (2-x)-\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{2+x} \, dx,x,\sqrt [3]{-8+(-2+x)^2}\right )+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{4-2 x+x^2} \, dx,x,\sqrt [3]{-8+(-2+x)^2}\right )\\ &=-\frac {3}{8} \log \left (2+\sqrt [3]{-8+(-2+x)^2}\right )+\frac {1}{4} \log (2-x)-\frac {3}{2} \operatorname {Subst}\left (\int \frac {1}{-12-x^2} \, dx,x,-2+2 \sqrt [3]{-8+(-2+x)^2}\right )\\ &=-\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1-\sqrt [3]{-8+(-2+x)^2}}{\sqrt {3}}\right )-\frac {3}{8} \log \left (2+\sqrt [3]{-8+(-2+x)^2}\right )+\frac {1}{4} \log (2-x)\\ \end {align*}
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Mathematica [C] time = 0.01, size = 36, normalized size = 0.41 \begin {gather*} \frac {3}{32} \left ((x-2)^2-8\right )^{2/3} \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {1}{8} \left (8-(x-2)^2\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 88, normalized size = 1.00 \begin {gather*} -\frac {1}{4} \log \left (\sqrt [3]{x^2-4 x-4}+2\right )+\frac {1}{8} \log \left (\left (x^2-4 x-4\right )^{2/3}-2 \sqrt [3]{x^2-4 x-4}+4\right )-\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {1}{\sqrt {3}}-\frac {\sqrt [3]{x^2-4 x-4}}{\sqrt {3}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 70, normalized size = 0.80 \begin {gather*} \frac {1}{4} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (x^{2} - 4 \, x - 4\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {1}{8} \, \log \left ({\left (x^{2} - 4 \, x - 4\right )}^{\frac {2}{3}} - 2 \, {\left (x^{2} - 4 \, x - 4\right )}^{\frac {1}{3}} + 4\right ) - \frac {1}{4} \, \log \left ({\left (x^{2} - 4 \, x - 4\right )}^{\frac {1}{3}} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{2} - 4 \, x - 4\right )}^{\frac {1}{3}} {\left (x - 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.98, size = 522, normalized size = 5.93 \begin {gather*} \frac {\ln \left (\frac {20 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{2}-80 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x +48 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-4 x -4\right )^{\frac {2}{3}}-23 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{2}-96 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-4 x -4\right )^{\frac {1}{3}}+92 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x -15 \left (x^{2}-4 x -4\right )^{\frac {2}{3}}+6 x^{2}+140 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )+30 \left (x^{2}-4 x -4\right )^{\frac {1}{3}}-24 x -56}{\left (-2+x \right )^{2}}\right )}{4}-\frac {\ln \left (\frac {20 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{2}-80 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x +48 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-4 x -4\right )^{\frac {2}{3}}-23 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{2}-96 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-4 x -4\right )^{\frac {1}{3}}+92 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x -15 \left (x^{2}-4 x -4\right )^{\frac {2}{3}}+6 x^{2}+140 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )+30 \left (x^{2}-4 x -4\right )^{\frac {1}{3}}-24 x -56}{\left (-2+x \right )^{2}}\right ) \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )}{2}+\frac {\RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \ln \left (\frac {40 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x^{2}-160 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )^{2} x -96 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-4 x -4\right )^{\frac {2}{3}}+6 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x^{2}+192 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) \left (x^{2}-4 x -4\right )^{\frac {1}{3}}-24 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right ) x +18 \left (x^{2}-4 x -4\right )^{\frac {2}{3}}-x^{2}-280 \RootOf \left (4 \textit {\_Z}^{2}-2 \textit {\_Z} +1\right )-36 \left (x^{2}-4 x -4\right )^{\frac {1}{3}}+4 x +28}{\left (-2+x \right )^{2}}\right )}{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (x^{2} - 4 \, x - 4\right )}^{\frac {1}{3}} {\left (x - 2\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (x-2\right )\,{\left (x^2-4\,x-4\right )}^{1/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (x - 2\right ) \sqrt [3]{x^{2} - 4 x - 4}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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